9. Point mode plotting with more than two images 2 hours

Transcription

1 Lecure // Cocep Hell/feiffer Februar 9. oi mode ploig wih more ha wo images hours aim: iersecio of more ha wo ras wih orieaed images Theor: Applicaio co lieari equaio 9.. Spaial Resecio ad Iersecio i hoogrammer The discussed procedures for ouer orieaio of phoogrammeric images, based o oe sep or wo seps procedures could be elarged o geeral case, whe more he wo images are ake ad ca be used for objec recosrucio. The procedure of sigle image space resecio ca be used ol i cases of moo ploig. Such cases usuall require addiioal iformaio abou he shape of he objecs. The have heir implemeaio ol for image recificaio or i cases of plae objecs. For wo images are possible hree approaches: - idepede resecio for ever phoo wihou akig io accou he ie pois. - simulaeous resecio for all images wih calculaio of ie pois coordiaes; - creaio of phoogrammeric model wih akig io accou of all poi or ol he par of hem. Orieaio of he model usig pois wih kow objec coordiaes. This approach leads o ver specific resuls. If all pois are used for relaive orieaio of he model he he adjused values of image coordiaes ca be obaied. If adjused values are used for calculaio of spaial coordiaes he i does o maer from which phoos hese image coordiaes are used because he resuls will be he same. If image coordiaes are o adjused or some image pois are o icluded i he adjusme he he space coordiaes of he poies mus be averaged or adjused. I mus be poied ha i his case i he process of ouer orieaio he relaive orieaio parameers cao be ake io accou ad sa fixed. B his reaso space coordiaes of objec pois cao ifluece he relaive orieaio parameers. As a fial resul he adjusme is o eirel correc. This approach ca be elarged for he cases of more ha wo phoos. I is made i mehods of srip ad block adjusme b mehod of models. Separae models are creaed ad afer ha are rasformed relaivel oe o aoher o he base of commo poi. I is possible o coec models b sequeial aachme of he ex phoo o he previous oe. These approaches will be discussed i more deails whe block adjusme is explaied. FH-KA - Maser course hoogrammer

2 Lecure // This approach is possible o be used i he case of close rage phoogrammer bu ver ofe he orieaio of he cameras is so differe ha i is o possible o creae sereo model. I his case ol several pois i he adjace phoos are measured i boh images. Tie pois are o eough o calculae parameers of relaive orieaio. This requires all phoos o be calculaed ogeher for deermiaio of orieaio parameers. Usuall he colieari codiios are used for his purpose. Such mehod of adjusme is called budle adjusme. All his pequliari require differe approach whe close rage images of he whole objec are processed for creaio he model of he whole objec. I he cases of close rage phoogrammer he orieaio of cameras differ so much ha is ecessar o fid good approximaio of his orieaio. 9.. Iiial approximaio for budle riagulaio Two problems are impora here: -approximae value for camera posiio; -alerae mehods for space resecio ad iersecio Approximae values for camera posiio ad ew pois To calculae approximae values for camera posiio i is ecessar o kow approximae values of agle orieaios of he camera. The ca be measured i he case of close rage phoogrammer whe he coordiae ssem is local ad usuall is orieed relaivel o he mai axes of he objec. The coefficies of roaio marix ca be calculaed from he coordiaes of ui vecors of image coordiae ssem. Ideed he relaio bewee image coordiaes ad space coordiaes is give b rasformaio equaio. x ξ r r r ξ Rωϕκ. η r r r. η z ζ r r r ζ where for simplificaio we deoe ξ η. If we subsiue for he vecors i camera coordiae ssem he ui vecors we obai: for vecor i, i zi r r r r (9.) xi r r r r i r r r. r (9.) FH-KA - Maser course hoogrammer

3 Lecure // for j vecor ad for k vecor. xi r r r r j r r r. r i zi r r r r xi r r r r k r r r. r i zi r r r r If we have arbirar orieed phoos i space we ca calculae roaio marixes from ui vecors. (9.) (9.4) FH-KA - Maser course hoogrammer

4 Lecure // z x O η z O x η ν ξ η ξ O x z ξ ν ν 4 Z κ O O κ O κ Y X Figure 9.. Three phoographs wih hree corol pois The calculaio of elemes of roaio marixes allow o solve he iiial approximaio for projecio ceers posiios ad coordiaes of ew pois from liear ssem of equaios. From colieari codiios makig assumpio for x we derive FH-KA - Maser course hoogrammer

5 Lecure // r.( X X) + r.( Y Y ) + r.( Z Z) x c r.( X X ) + r.( Y Y ) + r.( Z Z ) r.( X X) + r.( Y Y ) + r.( Z Z) c r.( X X ) + r.( Y Y ) + r.( Z Z ) We mulipl he lef par of equaios b deomiaor x.[ r.( X X ) + r.( Y Y ) + r.( Z Z )] c.[ r.( X X ) + r.( Y Y ) + r.( Z Z )].[ r.( X X ) + r.( Y Y ) + r.( Z Z )] c.[ r.( X X ) + r.( Y Y ) + r.( Z Z)] Afer rearragig he erms b parameers we obai he resul (. cr + xr. ). X + (. cr + xr. ). Y + (. cr + xr. ). Z ( cr. + xr. ). X ( cr. + xr. ). Y ( cr. + xr. ). Z (. cr + xr. ). X + (. cr + xr. ). Y + (. cr + xr. ). Z (. cr + xr. ). X (. cr + xr. ). Y (. cr + xr. ). Z I he above equaios here are six ukows coordiaes of projecio ceer ad coordiae of he objec poi (9.5) (9.6) (9.7) ( X, Y, Z ) ( X, YZ, ). I case whe his is a corol poi is coordiaes are give ad he are added o he lis of ukows. I complicaed ses of phoos usuall here are eough measuremes o differe pois ad i is possible o calculae all parameers. The umber of pois for calculaio of orieaio is made i Appedix. c c c p p p p p p. k (4. ) + (6. ) +... (9.8) k p where is umber of phoos ad is he umber of ew pois p I case whe umber of equaios is greaer he he umber of required ol he ecessar umber is ake. Whe he umber of equaios is more he required i is possible o appl leas square adjusme mehod o fid he soluio Aleraig resecio ad iersecio If he elemes of ouer orieaio are kow he i is possible o fid he coordiaes of pois b liear equaio where erms coaiig ( X, Y, Z ) This leads o followig expressio for fidig he coordiaes: (. cr + xr. ). X + (. cr + xr. ). Y+ (. cr + xr. ). Z are rasfer o he righ par of equaios. ( cr. + xr. ). X + ( cr. + xr. ). Y + ( cr. + xr. ). Z (9.9) (. cr + xr. ). X + (. cr + xr. ). Y+ (. cr + xr. ). Z ( cr. + x. r). X + ( cr. + xr. ). Y + ( cr. + xr. ). Z FH-KA - Maser course hoogrammer

6 Lecure // Oe mehod for liear space resecio is suggesed b Müller/Killia. This mehod is based o usage of four full corol pois. This is wih oe poi more ha ha is ecessar for a spaial resecio wih meric camera. Reduda iformaio is o used ol for improveme of accurac bu also for liearisaio of o-liear problem. O γ α β r r r c a b Figure 9.. Spaial resecio wih hree pois The kow values are coordiaes of pois. Agles o he pois ca be measured b image coordiaes ad ier orieaio parameers. The required values are coordiaes of projecio ceer of he camera ( X, Y, Z ). To solve he problem as he firs sep he leghs of erahedro mus be compued. The followig equaios ca be formulaed: r + r. r. r.cosα a r + r. r. r.cos β b +...cosγ r r r r c (9.) Afer rasformaio he fourh-degree equaio is formulaed. The form of his equaio is show i Appedix FH-KA - Maser course hoogrammer

7 Lecure // The soluio of fourh degree equaio ca be foud i differe was. Oe ver effecive suggesio is made b Killia. A fourh poi 4 is iroduced. I is possible o formulae secod fourh-degree equaio for variable v from pois, ad 4. Afer elimiaio procedure a liear equaio for v is reached. Thus he o-liear problem is rasformed o liear ad solved. Afer fidig he radii he coordiaes of projecio ceer ca be foud b iersecio of hree spheres wih radii r, r ad r (spaial arc iersecio). This also leads o fourh-degree equaio. The followig soluio of he problem is possible [followig Kraus, K., 997]. We projec he cross poi N of perpedicular from projecio ceer O wih hree pois plae (,, ). O r δ G c δ δ δ N b H a Figure 9.. Iersecio of projecio vecors o hree pois Three equaios are formulaed for calculaio he icremes of vecor O are formulaed. FH-KA - Maser course hoogrammer

8 Lecure // The plae orhogoal o a ad passig rough O is give b he scalar muliplicaio of wo vecors a ad r. Accordigl o his we ca cosruc he plae hrough O, ha is orhogoal o c. The values of agles i he surroudig plaes is calculaed b he leghs of correspodig edges. For plae ( O) hese edges are a, r, r. For plae (,, O) hese edges are b, r, r. The hird equaio for coordiaes of poi N is derived from he fac ha his poi lies i he plae of (,, ). I is defied b orhogoal vecor o wo of he vecors a ad b. The soluio of hree equaios for coordiae differeces XN, YN ad Z N allows o calculae he coordiaes X N, Y N, Z N X X + X N N N Y Y + Y N Z Z + Z N N (9.) The legh of perpediclar NO is calculaed b he relaio ( N N N) NO r N r X + Y + Z (9.4) Fiall he coordiaes of poi O are calculaed b he relaio X XN Y Y N + NO. Z Z N (9.5) I he soluio of spaial resecio wih hree kow pois he six elemes of ouer orieaio ca be calculaed from six liearised equaios. The ssem of equaios ca be sigular or illcodiioed if hree poi lie o he circular clider wih he projecio ceer. I he case of four pois here is o dagerous clider. Bu i his case he dagerous cosrucio is whe four pois lie o horoper curve, which lies o dagerous clider Roaio of budle of ras Afer calculaio of he projecio ceers posisio he ex sep is deermiaio of roaio bewee objec coordiae ssem ad image (camera) coordiae ssem. We mus fid he roaio marix R, ha rasforms image coordiaes x io objec X (omiig scalig ad raslaio). X Rx. (9.6) FH-KA - Maser course hoogrammer

9 Lecure // To solve he problem i is possible o defie he image ad objec coordiae ssems, coeced wih hree give pois. z O z O k η j k j η i p p ξ Z x ξ i x Y X 9.4.Ui vecors i objec ad image coordiae ssems The calculaio of ui vecors i image coordiae ssem is made o he base of measurig coordiaes of image pois. We form ( ξ, η, c), ( ξ, η, c) ad ( ξ, η, c). From hese vecors we calculae ui vecors i image coordiae ssem - i, j, k. The calculaio of ui vecors i objec coordiae ssem is more complicaed. For hese purpose i is ecessar o calculae he leghs of vecors o image pois. ' i i i O x + + c (9.7) Deailed calculaio of hese vecors is preseed i Appedix 4. Two ses of vecors defie roaio marixes. R i j k X R.x ' (9.8) FH-KA - Maser course hoogrammer

10 Lecure // Marix R produces ol roaio i he plae of he phoo R i j k x R. x ' (9.9) Combiig he vecor equaios we fid he oal roaio marix X RR.. x Rx. (9.) The preseaio of rasformaio marix is R RR. (9.) This approach allows o deermie more accurae he orieaio of image i objec coordiae ssem. However i requires hree corol pois (wih kow coordiaes o be measured i each phoo. Some imes such procedure is possible o be applied o o separae phoo bu for he whole phoogrammeric model. I such case he calculaios of models mus be doe ad relaive orieaio of models o be performed oo Sequece for resecio ad iersecio The procedure for deermiaio of iiial approximaio rus i followig sequece:.numberig of all phoographs.numberig of all pois i he se of phoograph.measureme of image coordiaes of all pois i all phoographs 4.Sorage of objec coordiaes of corol pois i a file 5.Trasfer he corol pois o he file of objec pois 6.Fidig phoographs, i which a leas four objec pois wih kow coordiaes are imaged 7.Spaial resecio of hese phoographs 8.Spaial iersecio for all ew pois ha lie i a leas wo phoographs for which he spaial resecio has bee calculaed 9.Trasfer objec coordiaes of calculaed pois o he file of objec pois.reur o sep 6 if here are o-orieed phoos..afer calculaio of all pois he procedures sops I some programs i is possible o use relaive orieaio for geeraio of models from phoos where here are o eough pois wih kow coordiaes Iiial approximaio for spaial rasformaio There are some case more ofe i close rage phoogrammer whe here are daa for spaial D model. I ca be for example: FH-KA - Maser course hoogrammer

11 Lecure // relaivel orieed phoogrammeric models; digial objec models (auomobile bod, buildig model from plas) ec. achmeer model i local Caresia coordiaes from polar measuremes; feaures defied i hree dimesioal coordiae ssem; projecio ceers wihi fligh srips, deermied b meas of GS measuremes; The calculaio is based o usage of equaio of spaial similari rasformaio X X + mr. x ΩΦΚ X X a a a x Y Y + a a a. Z Z a a a z (9.) Space similari rasformaio is defied b 7 parameers coordiaes of he ceer of coordiae ssem, scale ad hree roaio agles. This approach leads o o-lieari soluio. If for orieaio are used four homologues pois wih correspodig coordiaes i is possible o calculae he ukows of rasformaio. Afer ha he scale facor ca be compued from relaio m i j a ij Whe he scalig facor is calculaed all coefficies of roaioal marix are divided o m. aij rij for i, j m From he relaios for coefficies ad agles he agles values ca be calculaed. 4 b. x + b. x + b. x.z (9.) (9.4) Excep he commo pois i is possible o use commo sraigh lies or spaial lies o be used for ie elemes bewee images. The aalical formulas for hese lies are used ad heir parameers are added as ukows o he ormal ssem of equaios. I some cases he lie ca be represeed as iersecio of wo polomial surfaces. z c.. x 4 (9.5) 9..Soluio wih colieari codiios Whe he cofiguraio of phoos is arbirar ad here are more he wo measuremes i images for image poi he mos appropriae mehod is based o usage of colieari equaios. FH-KA - Maser course hoogrammer

12 Lecure // r.( X X) + r.( Y Y ) + r.( Z Z) x xp c r.( X X ) + r.( Y Y ) + r.( Z Z ) r.( X X) + r.( Y Y ) + r.( Z Z) p c r.( X X ) + r.( Y Y ) + r.( Z Z ) (9.6) The colieari equaios are observaio equaios because he calculaed values are direcl measured coordiaes. From hem he correcio equaios are produced r.( X X) + r.( Y Y ) + r.( Z Z) vx + xm xp c r.( X X ) + r.( Y Y ) + r.( Z Z ) r.( X X) + r.( Y Y ) + r.( Z Z) v + m p c r.( X X ) + r.( Y Y ) + r.( Z Z ) The ca be rasformed io he form r.( X X ) + r.( Y Y ) + r.( Z Z ) v x c x x p m r.( X X) + r.( Y Y ) + r.( Z Z) r.( X X) + r.( Y Y ) + r.( Z Z) v p c r.( X X ) + r.( Y Y ) + r.( Z Z ) m (9.7) (9.8) The correcio equaios are o-liear. The ca be liearised b differeiaig accordigl o he ukows v b δ X + b. δy + b δ Z + b δω + b δϕ + b δκ x b δx + b δy + b δz + b δx + b δc l 7 8 9, p, v b δ X + b. δy + b δ Z + b δω + b δϕ + b δκ b δx + b δy + b δz + b δ + b δc l 7 8 9, p, x (9.9) where l x x l x m m (9.) I marix form he correcio equaios are preseed i he form V B. x l (9.) ( m ) ( m ) ( ) ( m ) where m umber of observaios, umber of ukows. The leas square adjusme ca be applied if umber of observaio is greaer he he umber of ukows m>. The codiios for correc applicaio of leas square mehod are: a) he ormal disribuio of residuals vi wih zero mea value ad kow variace σ; b) muuall idepede observaios FH-KA - Maser course hoogrammer

13 Lecure // Formig ad soluio of ormal equaio ssem is give i Appedix 5. The soluio of ormal ssem is give b equaio x N L (9.) The variace-covariace marix for x is obaied b σx σ N (9.) The applicaio of colieari equaios ad adjusme b observaio equaios gives he soluios i all cases of combiaios of measuremes ad phoos. I is ecessar ha he umber of observaios o ew pois o be greaer or equal of wo. Appedixes Appedix The ecessar umber of equaios k is calculaed b he relaio k. k +. p p (9.4) k p where is umber of phoos ad is he umber of ew pois p The possible umber of equaios is defied b formula k [ + ( ) + ( ) + e p p p p p p p p p mmax ( ) +...]. m. p p p p pm m (9.5) where m is he umber of occurrece of poi ad pm is he umber of pois wih occurrece of order m. I is evide ha if here are ol pois image ol i oe phoo i is o possible o fid he soluio. Bu if pois are measured i wo or more phoos he i is possible o selec eough equaios. The umber of equaios from ew pois from which we ca deermie projecio ceers is decreased wih hree. So he followig relaio mus be saisfied. mmax c p + p pm + pm m. k.. m.( ) mmax c p pm pm m p. k [. m. + (. m.. )] The above equaio ca be preseed i he followig form: (9.6) c c c p p p p p p. k (4. ) + (6. ) +... (9.7) FH-KA - Maser course hoogrammer

14 Lecure // Appedix The followig ew ukow is iroduced. r v (9.8) r Fiall he equaio for v has he form 4 [. A+ B+ 4. C.si β ]. v α γ β v + 4.[ cos β.cos γ + A.(cos β.cosγ + cos α) B.cosα. C.si β.cos α]. v + [ + (cos β si γ) A( + cos βcosαcos γ) + B( + cos ) + ( b si + c si )] a + 4.[ cos β.cos γ + A.(cos β.cosγ + cos α) B.cosα. D.si γ.cos α]. v [. A+ B+ 4. D.si γ ] where A, B, C ad D are represeed b followig expressios (9.9) b c b c + c b A, B, C, D (9.4) a a a a Appedix The plae orhogoal o a ad passig rough O is give b he scalar muliplicaio of wo vecors a ad r. This gives he relaio a. r X. X + Y. Y + Z. Z. r.cos δ ar..cosδ H (9.4) where icremes of coordiaes are relaivel o poi. For he foo poi N, ha lies i he same plae we have he equaio ' N N N δ X. X + Y. Y + Z. Z. N.cos H ar..cosδ (9.4) Accordigl o his we ca cosruc he plae hrough O, ha is orhogoal o c. c. r cos δ..co δ (9.4) X X Y Y Z Z r cr s The equaio for he poi N i his plae akes he form ' N N N δ X. X + Y. Y + Z. Z. N.cos G cr..cosδ (9.44) FH-KA - Maser course hoogrammer

15 Lecure // I he above equaios he values of agles i he surroudig plaes is calculaed b he leghs of correspodig edges. For plae (O) hese edges are a, r, r. For plae (,, O) hese edges are b, r, r. The hird equaio for coordiaes of poi N is derived from he fac ha his poi lies i he plae of (,, ). I is defied b orhogoal vecor o wo of he vecors a ad b. a b x i j k a. bz az. b ax a az az. bx ax. bz z b x b b z ax. b a. b x The equaio for coordiaes of N i he plae of hree pois is (9.45). X +. Y +. Z (9.46) x N N z N The soluio of hree equaios for coordiae differeces XN, YN ad Z N allows o calculae he coordiaes X N, Y N, Z N Appedix 4 ' ' The ui vecor i is defied o lie o he lie,. ' ' ' ' i (9.47) Vecor k is orhogoal o he image plae, so we have k ' ' ' ' ' ' ' ' (9.48) Fiall i is possible o fid vecor j usig he fac ha we have orhogoal coordiae ssem. j k i (9.49) I he similar wa i is possible o calculae ui vecors i he image coordiae ssem. Two ses of vecors defie roaio marixes. Appedix 5 The ormal equaio ssem is formed b relaios ( ) BWB x BWl (9.5) FH-KA - Maser course hoogrammer

16 Lecure // If defie N BWB L BW l (9.5) The ormal ssem is preseed as Nx. L (9.5) The marix W is weigh marix ad i he case ucorrelaed measureme has he form W δ x σ x σ (9.5) σ x The soluio of ormal ssem is give b equaio m x N L (9.54) The variace-covariace marix for x is obaied b σx σ N (9.55) where σ is ui weigh variace ad i is calculaed b σ m j v j m (9.56) FH-KA - Maser course hoogrammer